
where \(c_2\)  is a positive constant of the order of \(2\sigma ^2\)  and \(H\)  is the local Hölder exponent. In a turbulent context, it is universally observed that \(H\approx 1/3\) , as predicted by dimensional arguments [1]}, [2]}, and Eq. REF  says that, at this second order statistical level, velocity shares the same local regularity as a fractional Brownian motion of Hurst parameter \(H\)  [3]}. Moreover, as a more precise characterization of the observed non-Gaussian nature of the velocity field, higher-order structure functions, i.e. the moments of order \(q\in \mathbb {N}\)  of the increments, behave as
\(\lim _{\nu \rightarrow 0}\mathbb {E}\left( \delta _\ell u_i\right)^q\mathrel {\mathop {\hspace{0.0pt}\sim }\limits _{\ell \rightarrow 0}^{}} c_q \left( \frac{\ell }{L}\right)^{\zeta _q},\) 
